Chapter XXI: Act 1871: , the jurisdiction, duties and command exercised by the lord (3)

The undulatory vibration postulated by Fresnel having been generally accepted as explaining most optical phenomena, it became necessary to determine the mechanical properties of the aether which transmits this motion. Fresnel, Neumann, Cauchy, MacCullagh, and, especially, Green and Stokes, developed the "elastic-solid theory." By applying the theory of elasticity they endeavoured to determine the constants of a medium which could transmit waves of the nature of light. Many different allocations were suggested (of which one of the most recent is Lord Kelvin's "contractile aether," which, however, was afterwards discarded by its author), and the theory as left by Green and Stokes has merits other than purely historical. At a later date theories involving an action between the aether and material atoms were proposed, the first of any moment being J. Boussinesq's (1867). C. Christiansen's investigation of anomalous dispersion in 1870, and the failure of Cauchy's formula (founded on the elastic-solid theory) to explain this phenomenon, led to the theories of W. Sellmeier (1872), H. von Helmholtz (1875), E. Ketteler (1878), E. Lommel (1878) and W. Voigt (1883). A third class of theory, to which the present-day theory belongs, followed from Clerk Maxwell's analytical investigations in electromagnetics. Of the greatest exponents of this theory we may mention H. A. Lorentz, P. Drude and J. Larmor, while Lord Rayleigh has, with conspicuous brilliancy, explained several phenomena (e.g. the colour of the sky) on this hypothesis.

For a critical examination of these theories see section II. of this
article; reference may also be made to the _British Association
Reports_: "On Physical Optics," by Humphrey Lloyd (1834), p. 35; "On
Double Refraction," by Sir G. G. Stokes (1862), p. 253; "On Optical
Theories," by R. T. Glazebrook (1885), p. 157.

§ 13. _Recent Developments._--The determination of the velocity of light (see section III. of this article) may be regarded as definitely settled, a result contributed to by A. H. L. Fizeau (1849), J. B. L. Foucault (1850, 1862), A. Cornu (1874), A. A. Michelson (1880), James Young and George Forbes (1882), Simon Newcomb (1880-1882) and Cornu (1900). The velocity in moving media was investigated theoretically by Fresnel; and Fizeau (1859), and Michelson and Morley (1886) showed experimentally that the velocity was increased in running water by an amount agreeing with Fresnel's formula, which was based on the hypothesis of a stationary aether. The optics of moving media have also been investigated by Lord Rayleigh, and more especially by H. A. Lorentz, who also assumed a stationary aether. The relative motion of the earth and the aether has an important connexion with the phenomenon of the aberration of light, and has been treated with masterly skill by Joseph Larmor and others (see AETHER). The relation of the earth's motion to the intensities of terrestrial sources of light was investigated theoretically by Fizeau, but no experimental inquiry was made until 1903, when Nordmeyer obtained negative results, which were confirmed by the theoretical investigations of A. A. Bucherer and H. A. Lorentz.

Experimental photometry has been greatly developed since the pioneer work of Bouguer and Lambert and the subsequent introduction of the photometers of Ritchie, Rumford, Bunsen and Wheatstone, followed by Swan's in 1859, and O. R. Lummer and E. Brodhun's instrument (essentially the same as Swan's) in 1889. This expansion may largely be attributed to the increase in the number of artificial illuminants--especially the many types of filament- and arc-electric lights, and the incandescent gas light. Colour photometry has also been notably developed, especially since the enunciation of the "Purkinje phenomenon" in 1825. Sir William Abney has contributed much to this subject, and A. M. Meyer has designed a photometer in which advantage is taken of the phenomenon of contrast colours. "Flicker photometry" may be dated from O. N. Rood's investigations in 1893, and the same principle has been applied by Haycraft and Whitman. These questions--colour and flicker photometry--have important affinities to colour perception and the persistence of vision (see VISION). The spectrophotometer, devised by De Witt Bristol Brace in 1899, which permits the comparison of similarly coloured portions of the spectra from two different sources, has done much valuable work in the determination of absorptive powers and extinction coefficients. Much attention has also been given to the preparation of a standard of intensity, and many different sources have been introduced (see PHOTOMETRY). Stellar photometry, which was first investigated instrumentally with success by Sir John Herschel, was greatly improved by the introduction of Zöllner's photometer, E. C. Pickering's meridian photometer and C. Pritchard's wedge photometer. Other methods of research in this field are by photography--photographic photometry--and radiometric method (see PHOTOMETRY, CELESTIAL).

The earlier methods for the experimental determination of refractive indices by measuring the deviation through a solid prism of the substance in question or, in the case of liquids, through a hollow prism containing the liquid, have been replaced in most accurate work by other methods. The method of total reflection, due originally to Wollaston, has been put into a very convenient form, applicable to both solids and liquids, in the Pulfrich refractometer (see REFRACTION). Still more accurate methods, based on interference phenomena, have been devised. Jamin's interference refractometer is one of the earlier forms of such apparatus; and Michelson's interferometer is one of the best of later types (see INTERFERENCE). The variation of refractive index with density has been the subject of much experimental and theoretical inquiry. The empirical rule of Gladstone and Dale was often at variance with experiment, and the mathematical investigations of H. A. Lorentz of Leiden and L. Lorenz of Copenhagen on the electromagnetic theory led to a more consistent formula. The experimental work has been chiefly associated with the names of H. H. Landolt and J. W. Brühl, whose results, in addition to verifying the Lorenz-Lorentz formula, have established that this function of the refractive index and density is a colligative property of the molecule, i.e. it is calculable additively from the values of this function for the component atoms, allowance being made for the mode in which they are mutually combined (see CHEMISTRY, PHYSICAL). The preparation of lenses, in which the refractive index decreases with the distance from the axis, by K. F. J. Exner, H. F. L. Matthiessen and Schott, and the curious results of refraction by non-homogeneous media, as realized by R. Wood may be mentioned (see MIRAGE).

The spectrum of white light produced by prismatic refraction has engaged many investigators. The infra-red or heat waves were discovered by Sir William Herschel, and experiments on the actinic effects of the different parts of the spectrum on silver salts by Scheele, Senebier, Ritter, Seebeck and others, proved the increased activity as one passed from the red to the violet and the ultra-violet. Wollaston also made many investigations in this field, noticing the dark lines--the "Fraunhofer lines"--which cross the solar spectrum, which were further discussed by Brewster and Fraunhofer, who thereby laid the foundations of modern spectroscopy. Mention may also be made of the investigations of Lord Rayleigh and Arthur Schuster on the resolving power of prisms (see DIFFRACTION), and also of the modern view of the function of the prism in analysing white light. The infra-red and ultra-violet rays are of especial interest since, although not affecting vision after the manner of ordinary light, they possess very remarkable properties. Theoretical investigation on the undulatory theory of the law of reflection shows that a surface, too rough to give any trace of regular reflection with ordinary light, may regularly reflect the long waves, a phenomenon experimentally realized by Lord Rayleigh. Long waves--the so-called "residual rays" or "_Rest-strahlen_"--have also been isolated by repeated reflections from quartz surfaces of the light from zirconia raised to incandescence by the oxyhydrogen flame (E. F. Nichols and H. Rubens); far longer waves were isolated by similar reflections from fluorite (56 µ) and sylvite (61 µ) surfaces in 1899 by Rubens and E. Aschkinass. The short waves--ultra-violet rays--have also been studied, the researches of E. F. Nichols on the transparency of quartz to these rays, which are especially present in the radiations of the mercury arc, having led to the introduction of lamps made of fused quartz, thus permitting the convenient study of these rays, which, it is to be noted, are absorbed by ordinary clear glass. Recent researches at the works of Schott and Genossen, Jena, however, have resulted in the production of a glass transparent to the ultra-violet.

Dispersion, i.e. that property of a substance which consists in having a different refractive index for rays of different wave-lengths, was first studied in the form known as "ordinary dispersion" in which the refrangibility of the ray increased with the wave-length. Cases had been observed by Fox Talbot, Le Roux, and especially by Christiansen (1870) and A. Kundt (1871-1872) where this normal rule did not hold; to such phenomena the name "anomalous dispersion" was given, but really there is nothing anomalous about it at all, ordinary dispersion being merely a particular case of the general phenomenon. The Cauchy formula, which was founded on the elastic-solid theory, did not agree with the experimental facts, and the germs of the modern theory, as was pointed out by Lord Rayleigh in 1900, were embodied in a question proposed by Clerk Maxwell for the Mathematical Tripos examination for 1869. The principle, which occurred simultaneously to W. Sellmeier (who is regarded as the founder of the modern theory) and had been employed about 1850 by Sir G. G. Stokes to explain absorption lines, involves an action between the aether and the molecules of the dispersing substance. The mathematical investigation is associated with the names of Sellmeier, Hermann Helmholtz, Eduard Ketteler, P. Drude, H. A. Lorentz and Lord Rayleigh, and the experimental side with many observers--F. Paschen, Rubens and others; absorbing media have been investigated by A. W. Pflüger, a great many aniline dyes by K. Stöckl, and sodium vapour by R. W. Wood. Mention may also be made of the beautiful experiments of Christiansen (1884) and Lord Rayleigh on the colours transmitted by white powders suspended in liquids of the same refractive index. If, for instance, benzol be gradually added to finely powdered quartz, a succession of beautiful colours--red, yellow, green and finally blue--is transmitted, or, under certain conditions, the colours may appear at once, causing the mixture to flash like a fiery opal. Absorption, too, has received much attention; the theory has been especially elaborated by M. Planck, and the experimental investigation has been prosecuted from the purely physical standpoint, and also from the standpoint of the physical chemist, with a view to correlating absorption with constitution.

Interference phenomena have been assiduously studied. The experiments of Young, Fresnel, Lloyd, Fizeau and Foucault, of Fresnel and Arago on the measurement of refractive indices by the shift of the interference bands, of H. F. Talbot on the "Talbot bands" (which he insufficiently explained on the principle of interference, it being shown by Sir G. B. Airy that diffraction phenomena supervene), of Baden-Powell on the "Powell bands," of David Brewster on "Brewster's bands," have been developed, together with many other phenomena--Newton's rings, the colours of thin, thick and mixed plates, &c.--in a striking manner, one of the most important results being the construction of interferometers applicable to the determination of refractive indices and wave-lengths, with which the names of Jamin, Michelson, Fabry and Perot, and of Lummer and E. Gehrcke are chiefly associated. The mathematical investigations of Fresnel may be regarded as being completed by the analysis chiefly due to Airy, Stokes and Lord Rayleigh. Mention may be made of Sir G. G. Stokes' attribution of the colours of iridescent crystals to periodic twinning; this view has been confirmed by Lord Rayleigh (_Phil. Mag._, 1888) who, from the purity of the reflected light, concluded that the laminae were equidistant by the order of a wave-length. Prior to 1891 only interference between waves proceeding in the same direction had been studied. In that year Otto H. Wiener obtained, on a film 1/20th of a wave-length in thickness, photographic impressions of the stationary waves formed by the interference of waves proceeding in opposite directions, and in 1892 Drude and Nernst employed a fluorescent film to record the same phenomenon. This principle is applied in the Lippmann colour photography, which was suggested by W. Zenker, realized by Gabriel Lippmann, and further investigated by R. G. Neuhauss, O. H. Wiener, H. Lehmann and others.

Great progress has been made in the study of diffraction, and "this department of optics is precisely the one in which the wave theory has secured its greatest triumphs" (Lord Rayleigh). The mathematical investigations of Fresnel and Poisson were placed on a dynamical basis by Sir G. G. Stokes; and the results gained more ready interpretation by the introduction of "Babinet's principle" in 1837, and Cornu's graphic methods in 1874. The theory also gained by the researches of Fraunhofer, Airy, Schwerd, E. Lommel and others. The theory of the concave grating, which resulted from H. A. Rowland's classical methods of ruling lines of the necessary nature and number on curved surfaces, was worked out by Rowland, E. Mascart, C. Runge and others. The resolving power and the intensity of the spectra have been treated by Lord Rayleigh and Arthur Schuster, and more recently (1905), the distribution of light has been treated by A. B. Porter. The theory of diffraction is of great importance in designing optical instruments, the theory of which has been more especially treated by Ernst Abbe (whose theory of microscopic vision dates from about 1870) by the scientific staff at the Zeiss works, Jena, by Rayleigh and others. The theory of coronae (as diffraction phenomena) was originally due to Young, who, from the principle involved, devised the _eriometer_ for measuring the diameters of very small objects; and Sir G. G. Stokes subsequently explained the appearances presented by minute opaque particles borne on a transparent plate. The polarization of the light diffracted at a slit was noted in 1861 by Fizeau, whose researches were extended in 1892 by H. Du Bois, and, for the case of gratings, by Du Bois and Rubens in 1904. The diffraction of light by small particles was studied in the form of very fine chemical precipitates by John Tyndall, who noticed the polarization of the beautiful cerulean blue which was transmitted. This subject--one form of which is presented in the blue colour of the sky--has been most auspiciously treated by Lord Rayleigh on both the elastic-solid and electromagnetic theories. Mention may be made of R. W. Wood's experiments on thin metal films which, under certain conditions, originate colour phenomena inexplicable by interference and diffraction. These colours have been assigned to the principle of optical resonance, and have been treated by Kossonogov (_Phys. Zeit._, 1903). J. C. Maxwell Garnett (_Phil. Trans_. vol. 203) has shown that the colours of coloured glasses are due to ultra-microscopic particles, which have been directly studied by H. Siedentopf and R. Zsigmondy under limiting oblique illumination.

Polarization phenomena may, with great justification, be regarded as the most engrossing subject of optical research during the 19th century; the assiduity with which it was cultivated in the opening decades of that century received a great stimulus when James Nicol devised in 1828 the famous "Nicol prism," which greatly facilitated the determination of the plane of vibration of polarized light, and the facts that light is polarized by reflection, repeated refractions, double refraction and by diffraction also contributed to the interest which the subject excited. The rotation of the plane of polarization by quartz was discovered in 1811 by Arago; if white light be used the colours change as the Nicol rotates--a phenomenon termed by Biot "rotatory dispersion." Fresnel regarded rotatory polarization as compounded from right- and left-handed (dextro- and laevo-) circular polarizations; and Fresnel, Cornu, Dove and Cotton effected their experimental separation. Legrand des Cloizeaux discovered the enormously enhanced rotatory polarization of cinnabar, a property also possessed--but in a lesser degree--by the sulphates of strychnine and ethylene diamine. The rotatory power of certain liquids was discovered by Biot in 1815; and at a later date it was found that many solutions behaved similarly. A. Schuster distinguishes substances with regard to their action on polarized light as follows: substances which act in the isotropic state are termed _photogyric_; if the rotation be associated with crystal structure, _crystallogyric_; if the rotation be due to a magnetic field, _magnetogyric_; for cases not hitherto included the term _allogyric_ is employed, while optically inactive substances are called _isogyric_. The theory of photogyric and crystallogyric rotation has been worked out on the elastic-solid (MacCullagh and others) and on the electromagnetic hypotheses (P. Drude, Cotton, &c.). Allogyrism is due to a symmetry of the molecule, and is a subject of the greatest importance in modern (and, more especially, organic) chemistry (see STEREOISOMERISM).

The optical properties of metals have been the subject of much experimental and theoretical inquiry. The explanations of MacCullagh and Cauchy were followed by those of Beer, Eisenlohr, Lundquist, Ketteler and others; the refractive indices were determined both directly (by Kundt) and indirectly by means of Brewster's law; and the reflecting powers from [lambda] = 251 µµ to [lambda] = 1500 µµ were determined in 1900-1902 by Rubens and Hagen. The correlation of the optical and electrical constants of many metals has been especially studied by P. Drude (1900) and by Rubens and Hagen (1903).

The transformations of luminous radiations have also been studied. John Tyndall discovered calorescence. Fluorescence was treated by John Herschel in 1845, and by David Brewster in 1846, the theory being due to Sir G. G. Stokes (1852). More recent studies have been made by Lommel, E. L. Nichols and Merritt (_Phys. Rev._, 1904), and by Millikan who discovered polarized fluorescence in 1895. Our knowledge of phosphorescence was greatly improved by Becquerel, and Sir James Dewar obtained interesting results in the course of his low temperature researches (see LIQUID GASES). In the theoretical and experimental study of radiation enormous progress has been recorded. The pressure of radiation, the necessity of which was demonstrated by Clerk Maxwell on the electromagnetic theory, and, in a simpler manner, by Joseph Larmor in his article RADIATION in these volumes, has been experimentally determined by E. F. Nichols and Hull, and the tangential component by J. H. Poynting. With the theoretical and practical investigation the names of Balfour Stewart, Kirchhoff, Stefan, Bartoli, Boltzmann, W. Wien and Larmor are chiefly associated. Magneto-optics, too, has been greatly developed since Faraday's discovery of the rotation of the plane of polarization by the magnetic field. The rotation for many substances was measured by Sir William H. Perkin, who attempted a correlation between rotation and composition. Brace effected the analysis of the beam into its two circularly polarized components, and in 1904 Mills measured their velocities. The Kerr effect, discovered in 1877, and the Zeeman effect (1896) widened the field of research, which, from its intimate connexion with the nature of light and electromagnetics, has resulted in discoveries of the greatest importance.

§ 14. _Optical Instruments._--Important developments have been made in the construction and applications of optical instruments. To these three factors have contributed. The mathematician has quantitatively analysed the phenomena observed by the physicist, and has inductively shown what results are to be expected from certain optical systems. A consequence of this was the detailed study, and also the preparation, of glasses of diverse properties; to this the chemist largely contributed, and the manufacture of the so-called _optical glass_ (see GLASS) is possibly the most scientific department of glass manufacture. The mathematical investigations of lenses owe much to Gauss, Helmholtz and others, but far more to Abbe, who introduced the method of studying the aberrations separately, and applied his results with conspicuous skill to the construction of optical systems. The development of Abbe's methods constitutes the main subject of research of the present-day optician, and has brought about the production of telescopes, microscopes, photographic lenses and other optical apparatus to an unprecedented pitch of excellence. Great improvements have been effected in the stereoscope. Binocular instruments with enhanced stereoscopic vision, an effect achieved by increasing the distance between the object glasses, have been introduced. In the study of diffraction phenomena, which led to the technical preparation of gratings, the early attempts of Fraunhofer, Nobert and Lewis Morris Rutherfurd, were followed by H. A. Rowland's ruling of plane and concave gratings which revolutionized spectroscopic research, and, in 1898, by Michelson's invention of the echelon grating. Of great importance are interferometers, which permit extremely accurate determinations of refractive indices and wave-lengths, and Michelson, from his classical evaluation of the standard metre in terms of the wave-lengths of certain of the cadmium rays, has suggested the adoption of the wave-length of one such ray as a standard with which national standards of length should be compared. Polarization phenomena, and particularly the rotation of the plane of polarization by such substances as sugar solutions, have led to the invention and improvements of polarimeters. The polarized light employed in such instruments is invariably obtained by transmission through a fixed Nicol prism--the polarizer--and the deviation is measured by the rotation of a second Nicol--the analyser. The early forms, which were termed "light and shade" polarimeters, have been generally replaced by "half-shade" instruments. Mention may also be made of the microscopic examination of objects in polarized light, the importance of which as a method of crystallographic and petrological research was suggested by Nicol, developed by Sorby and greatly expanded by Zirkel, Rosenbusch and others.

BIBLIOGRAPHY.--There are numerous text-books which give elementary
expositions of light and optical phenomena. More advanced works, which
deal with the subject experimentally and mathematically, are A. B.
Bassett, _Treatise on Physical Optics_ (1892); Thomas Preston, _Theory
of Light_, 2nd ed. by C. F. Joly (1901); R. W. Wood, _Physical Optics_
(1905), which contains expositions on the electromagnetic theory, and
treats "dispersion" in great detail. Treatises more particularly
theoretical are James Walker, _Analytical Theory of Light_ (1904); A.
Schuster, _Theory of Optics_ (1904); P. Drude, _Theory of Optics_,
Eng. trans. by C. R. Mann and R. A. Millikan (1902). General treatises
of exceptional merit are A. Winkelmann, _Handbuch der Physik_, vol.
vi. "Optik" (1904); and E. Mascart, _Traité d'optique_ (1889-1893); M.
E. Verdet, _Leçons d'optique physique_ (1869, 1872) is also a valuable
work. Geometrical optics is treated in R. S. Heath, _Geometrical
Optics_ (2nd ed., 1898); H. A. Herman, _Treatise on Geometrical
Optics_ (1900). Applied optics, particularly with regard to the theory
of optical instruments, is treated in H. D. Taylor, _A System of
Applied Optics_ (1906); E. T. Whittaker, _The Theory of Optical
Instruments_ (1907); in the publications of the scientific staff of
the Zeiss works at Jena: _Die Theorie der optischen Instrumente_, vol.
i. "Die Bilderzeugung in optischen Instrumenten" (1904); in S.
Czapski, _Theorie der optischen Instrumente_, 2nd ed. by O. Eppenstein
(1904); and in A. Steinheil and E. Voit, _Handbuch der angewandten
Optik_ (1901). The mathematical theory of general optics receives
historical and modern treatment in the _Encyklopädie der
mathematischen Wissenschaften_ (Leipzig). Meteorological optics is
fully treated in J. Pernter, _Meteorologische Optik_; and
physiological optics in H. v Helmholtz, _Handbuch der physiologischen
Optik_ (1896) and in A. Koenig, _Gesammelte Abhandlungen zur
physiologischen Optik_ (1903).

The history of the subject may be studied in J. C. Poggendorff,
_Geschichte der Physik_ (1879); F. Rosenberger, _Die Geschichte der
Physik_ (1882-1890); E. Gerland and F. Traumüller, _Geschichte der
physikalischen Experimentierkunst_ (1899); reference may also be made
to Joseph Priestley, _History and Present State of Discoveries
relating to Vision, Light and Colours_ (1772), German translation by
G. S. Klügel (Leipzig, 1775). Original memoirs are available in many
cases in their author's "collected works," e.g. Huygens, Young,
Fresnel, Hamilton, Cauchy, Rowland, Clerk Maxwell, Stokes (and also
his _Burnett Lectures on Light_), Kelvin (and also his _Baltimore
Lectures_, 1904) and Lord Rayleigh. Newton's _Opticks_ forms volumes
96 and 97 of Ostwald's Klassiker; Huygens' _Über d. Licht_ (1678),
vol. 20, and Kepler's _Dioptrice_ (1611), vol. 144 of the same series.

Contemporary progress is reported in current scientific journals, e.g.
the _Transactions_ and _Proceedings_ of the Royal Society, and of the
Physical Society (London), the _Philosophical Magazine_ (London), the
_Physical Review_ (New York, 1893 seq.) and in the _British
Association Reports_; in the _Annales de chimie et de physique and
Journal de physique_ (Paris); and in the _Physikalische Zeitschrift_
(Leipzig) and the _Annalen der Physik und Chemie_ (since 1900:
_Annalen der Physik_) (Leipzig). (C. E.*)

II. NATURE OF LIGHT

1. _Newton's Corpuscular Theory._--Until the beginning of the 19th century physicists were divided between two different views concerning the nature of optical phenomena. According to the one, luminous bodies emit extremely small corpuscles which can freely pass through transparent substances and produce the sensation of light by their impact against the retina. This _emission_ or _corpuscular theory_ of light was supported by the authority of Isaac Newton,[8] and, though it has been entirely superseded by its rival, the _wave-theory_, it remains of considerable historical interest.

2. _Explanation of Reflection and Refraction._--Newton supposed the light-corpuscles to be subjected to attractive and repulsive forces exerted at very small distances by the particles of matter. In the interior of a homogeneous body a corpuscle moves in a straight line as it is equally acted on from all sides, but it changes its course at the boundary of two bodies, because, in a thin layer near the surface there is a resultant force in the direction of the normal. In modern language we may say that a corpuscle has at every point a definite potential energy, the value of which is constant throughout the interior of a homogeneous body, and is even equal in all bodies of the same kind, but changes from one substance to another. If, originally, while moving in air, the corpuscles had a definite velocity v0, their velocity v in the interior of any other substance is quite determinate. It is given by the equation ½mv² - ½mv0² = A, in which m denotes the mass of a corpuscle, and A the excess of its potential energy in air over that in the substance considered.

A ray of light falling on the surface of separation of two bodies is
reflected according to the well-known simple law, if the corpuscles
are acted on by a sufficiently large force directed towards the first
medium. On the contrary, whenever the field of force near the surface
is such that the corpuscles can penetrate into the interior of the
second body, the ray is refracted. In this case the law of Snellius
can be deduced from the consideration that the projection w of the
velocity on the surface of separation is not altered, either in
direction or in magnitude. This obviously requires that the plane
passing through the incident and the refracted rays be normal to the
surface, and that, if [alpha]1 and [alpha]2 are the angles of
incidence and of refraction, v1 and v2 the velocities of light in the
two media,

sin [alpha]1/sin [alpha]2 = w/v1 : w/v2 = v2/v1. (1)

The ratio is constant, because, as has already been observed, v1 and
v2 have definite values.

As to the unequal refrangibility of differently coloured light, Newton
accounted for it by imagining different kinds of corpuscles. He
further carefully examined the phenomenon of total reflection, and
described an interesting experiment connected with it. If one of the
faces of a glass prism receives on the inside a beam of light of such
obliquity that it is totally reflected under ordinary circumstances,
a marked change is observed when a second piece of glass is made to
approach the reflecting face, so as to be separated from it only by a
very thin layer of air. The reflection is then found no longer to be
total, part of the light finding its way into the second piece of
glass. Newton concluded from this that the corpuscles are attracted by
the glass even at a certain small measurable distance.

3. _New Hypotheses in the Corpuscular Theory._--The preceding explanation of reflection and refraction is open to a very serious objection. If the particles in a beam of light all moved with the same velocity and were acted on by the same forces, they all ought to follow exactly the same path. In order to understand that part of the incident light is reflected and part of it transmitted, Newton imagined that each corpuscle undergoes certain alternating changes; he assumed that in some of its different "phases" it is more apt to be reflected, and in others more apt to be transmitted. The same idea was applied by him to the phenomena presented by very thin layers. He had observed that a gradual increase of the thickness of a layer produces periodic changes in the intensity of the reflected light, and he very ingeniously explained these by his theory. It is clear that the intensity of the transmitted light will be a minimum if the corpuscles that have traversed the front surface of the layer, having reached that surface while in their phase of easy transmission, have passed to the opposite phase the moment they arrive at the back surface. As to the nature of the alternating phases, Newton (_Opticks_, 3rd ed., 1721, p. 347) expresses himself as follows:--"Nothing more is requisite for putting the Rays of Light into Fits of easy Reflexion and easy Transmission than that they be small Bodies which by their attractive Powers, or some other Force, stir up Vibrations in what they act upon, which Vibrations being swifter than the Rays, overtake them successively, and agitate them so as by turns to increase and decrease their Velocities, and thereby put them into those Fits."

4. _The Corpuscular Theory and the Wave-Theory compared._--Though Newton introduced the notion of periodic changes, which was to play so prominent a part in the later development of the wave-theory, he rejected this theory in the form in which it had been set forth shortly before by Christiaan Huygens in his _Traité de la lumière_ (1690), his chief objections being: (1) that the rectilinear propagation had not been satisfactorily accounted for; (2) that the motions of heavenly bodies show no sign of a resistance due to a medium filling all space; and (3) that Huygens had not sufficiently explained the peculiar properties of the rays produced by the double refraction in Iceland spar. In Newton's days these objections were of much weight.

Yet his own theory had many weaknesses. It explained the propagation in straight lines, but it could assign no cause for the equality of the speed of propagation of all rays. It adapted itself to a large variety of phenomena, even to that of double refraction (Newton says [ibid.]:--"... the unusual Refraction of Iceland Crystal looks very much as if it were perform'd by some kind of attractive virtue lodged in certain Sides both of the Rays, and of the Particles of the Crystal."), but it could do so only at the price of losing much of its original simplicity.

In the earlier part of the 19th century, the corpuscular theory broke down under the weight of experimental evidence, and it received the final blow when J. B. L. Foucault proved by direct experiment that the velocity of light in water is not greater than that in air, as it should be according to the formula (1), but less than it, as is required by the wave-theory.

5. _General Theorems on Rays of Light._--With the aid of suitable assumptions the Newtonian theory can accurately trace the course of a ray of light in any system of isotropic bodies, whether homogeneous or otherwise; the problem being equivalent to that of determining the motion of a material point in a space in which its potential energy is given as a function of the coordinates. The application of the dynamical principles of "least and of varying action" to this latter problem leads to the following important theorems which William Rowan Hamilton made the basis of his exhaustive treatment of systems of rays.[9] The total energy of a corpuscle is supposed to have a given value, so that, since the potential energy is considered as known at every point, the velocity v is so likewise.

(a) The path along which light travels from a point A to a point B is
determined by the condition that for this line the integral [int]v ds,
in which ds is an element of the line, be a minimum (provided A and B
be not too near each other). Therefore, since v = µv0, if v0 is the
velocity of light _in vacuo_ and µ the index of refraction, we have
for every variation of the path the points A and B remaining fixed,

[delta][int]µ ds = 0. (2)

(b) Let the point A be kept fixed, but let B undergo an infinitely
small displacement BB´ (=q) in a direction making an angle [theta]
with the last element of the ray AB. Then, comparing the new ray AB´
with the original one, it follows that

[delta][int]µ ds = µ_B q cos [theta], (3)

where µ_B is the value of µ at the point B.

6. _General Considerations on the Propagation of Waves._--"Waves," i.e. local disturbances of equilibrium travelling onward with a certain speed, can exist in a large variety of systems. In a theory of these phenomena, the state of things at a definite point may in general be defined by a certain directed or vector quantity P,[10] which is zero in the state of equilibrium, and may be called the disturbance (for example, the velocity of the air in the case of sound vibrations, or the displacement of the particles of an elastic body from their positions of equilibrium). The components P_x, P_y, P_z of the disturbance in the directions of the axes of coordinates are to be considered as functions of the coordinates x, y, z and the time t, determined by a set of partial differential equations, whose form depends on the nature of the problem considered. If the equations are homogeneous and linear, as they always are for sufficiently small disturbances, the following theorems hold.

(a) Values of P_x, P_y, P_z (expressed in terms of x, y, z, t) which
satisfy the equations will do so still after multiplication by a
common arbitrary constant.

(b) Two or more solutions of the equations may be combined into a new
solution by addition of the values of P_x, those of P_y, &c., i.e. by
compounding the vectors P, such as they are in each of the particular
solutions.

In the application to light, the first proposition means that the
phenomena of propagation, reflection, refraction, &c., can be produced
in the same way with strong as with weak light. The second proposition
contains the principle of the "superposition" of different states, on
which the explanation of all phenomena of interference is made to
depend.

In the simplest cases (monochromatic or homogeneous light) the
disturbance is a simple harmonic function of the time ("simple
harmonic vibrations"), so that its components can be represented by

P_x = a1 cos (nt + f1),
P_y = a2 cos (nt + f2),
P_z = a3 cos (nt + f3).

The "phases" of these vibrations are determined by the angles nt + f1,
&c., or by the times t + f1/n, &c. The "frequency" n is constant
throughout the system, while the quantities f1, f2, f3, and perhaps
the "amplitudes" a1, a2, a3 change from point to point. It may be
shown that the end of a straight line representing the vector P, and
drawn from the point considered, in general describes a certain
ellipse, which becomes a straight line, if f1 = f2 = f3. In this
latter case, to which the larger part of this article will be
confined, we can write in vector notation

P = A cos (nt + f), (4)

where A itself is to be regarded as a vector.

We have next to consider the way in which the disturbance changes from
point to point. The most important case is that of plane waves with
constant amplitude A. Here f is the same at all points of a plane
("wave-front") of a definite direction, but changes as a linear
function as we pass from one such wave-front to the next. The axis of
x being drawn at right angles to the wave-fronts, we may write f = f0
- kx, where f0 and k are constants, so that (4) becomes

P = A cos (nt - kx + f0). (5)

This expression has the period 2[pi]/n with respect to the time and
the perion 2[pi]/k with respect to x, so that the "time of vibration"
and the "wave-length" are given by T = 2[pi]/n, [lambda] = 2[pi]/k.
Further, it is easily seen that the phase belonging to certain values
of x and t is equal to that which corresponds to x + [Delta]x and t +
[Delta]t provided [Delta]x = (n/k)[Delta]t. Therefore the phase, or
the disturbance itself, may be said to be propagated in the direction
normal to the wave-fronts with a velocity (velocity of the waves) v =
n/k, which is connected with the time of vibration and the wave-length
by the relation

[lambda] = vT. (6)

In isotropic bodies the propagation can go on in all directions with
the same velocity. In anisotropic bodies (crystals), with which the
theory of light is largely concerned, the problem is more complicated.
As a general rule we can say that, for a given direction of the
wave-fronts, the vibrations must have a determinate direction, if the
propagation is to take place according to the simple formula given
above. It is to be understood that for a given direction of the waves
there may be two or even more directions of vibration of the kind, and
that in such a case there are as many different velocities, each
belonging to one particular direction of vibration.

7. _Wave-surface._--After having found the values of v for a particular frequency and different directions of the wave-normal, a very instructive graphical representation can be employed.

Let ON be a line in any direction, drawn from a fixed point O, OA a
length along this line equal to the velocity v of waves having ON for
their normal, or, more generally, OA, OA´, &c., lengths equal to the
velocities v, v´, &c., which such waves have according to their
direction of vibration, Q, Q´, &c., planes perpendicular to ON through
A, A^1, &c. Let this construction be repeated for all directions of
ON, and let W be the surface that is touched by all the planes Q, Q´,
&c. It is clear that if this surface, which is called the
"wave-surface," is known, the velocity of propagation of plane waves
of any chosen direction is given by the length of the perpendicular
from the centre O on a tangent plane in the given direction. It must
be kept in mind that, in general, each tangent plane corresponds to
one definite direction of vibration. If this direction is assigned in
each point of the wave-surface, the diagram contains all the
information which we can desire concerning the propagation of plane
waves of the frequency that has been chosen.

The plane Q employed in the above construction is the position after
unit of time of a wave-front perpendicular to ON and originally
passing through the point O. The surface W itself is often considered
as the locus of all points that are reached in unit of time by a
disturbance starting from O and spreading towards all sides. Admitting
the validity of this view, we can determine in a similar way the locus
of the points reached in some infinitely short time dt, the
wave-surface, as we may say, or the "elementary wave," corresponding
to this time. It is similar to W, all dimensions of the latter surface
being multiplied by dt. It may be noticed that in a heterogeneous
medium a wave of this kind has the same form as if the properties of
matter existing at its centre extended over a finite space.

8. _Theory of Huygens._--Huygens was the first to show that the explanation of optical phenomena may be made to depend on the wave-surface, not only in isotropic bodies, in which it has a spherical form, but also in crystals, for one of which (Iceland spar) he deduced the form of the surface from the observed double refraction. In his argument Huygens availed himself of the following principle that is justly named after him: Any point that is reached by a wave of light becomes a new centre of radiation from which the disturbance is propagated towards all sides. On this basis he determined the progress of light-waves by a construction which, under a restriction to be mentioned in § 13, applied to waves of any form and to all kinds of transparent media. Let [sigma] be the surface (wave-front) to which a definite phase of vibration has advanced at a certain time t, dt an infinitely small increment of time, and let an elementary wave corresponding to this interval be described around each point P of [sigma]. Then the envelope [sigma]´ of all these elementary waves is the surface reached by the phase in question at the time t + dt, and by repeating the construction all successive positions of the wave-front can be found.

Huygens also considered the propagation of waves that are laterally
limited, by having passed, for example, through an opening in an
opaque screen. If, in the first wave-front [sigma], the disturbance
exists only in a certain part bounded by the contour s, we can confine
ourselves to the elementary waves around the points of that part, and
to a portion of the new wave-front [sigma]´ whose boundary passes
through the points where [sigma]´ touches the elementary waves having
their centres on s. Taking for granted Huygens's assumption that a
sensible disturbance is only found in those places where the
elementary waves are touched by the new wave-front, it may be inferred
that the lateral limits of the beam of light are determined by lines,
each element of which joins the centre P of an elementary wave with
its point of contact P´ with the next wave-front. To lines of this
kind, whose course can be made visible by using narrow pencils of
light, the name of "rays" is to be given in the wave-theory. The
disturbance may be conceived to travel along them with a velocity u =
PP´/dt, which is therefore called the "ray-velocity."

The construction shows that, corresponding to each direction of the
wave-front (with a determinate direction of vibration), there is a
definite direction and a definite velocity of the ray. Both are given
by a line drawn from the centre of the wave-surface to its point of
contact with a tangent plane of the given direction. It will be
convenient to say that this line and the plane are conjugate with each
other. The rays of light, curved in non-homogeneous bodies, are always
straight lines in homogeneous substances. In an isotropic medium,
whether homogeneous or otherwise, they are normal to the wave-fronts,
and their velocity is equal to that of the waves.

By applying his construction to the reflection and refraction of
light, Huygens accounted for these phenomena in isotropic bodies as
well as in Iceland spar. It was afterwards shown by Augustin Fresnel
that the double refraction in biaxal crystals can be explained in the
same way, provided the proper form be assigned to the wave-surface.

In any point of a bounding surface the normals to the reflected and
refracted waves, whatever be their number, always lie in the plane
passing through the normal to the incident waves and that to the
surface itself. Moreover, if [alpha]1 is the angle between these two
latter normals, and [alpha]2 the angle between the normal to the
boundary and that to any one of the reflected and refracted waves, and
v1, v2 the corresponding wave-velocities, the relation

sin [alpha]1/sin [alpha]2 = v1/v2 (7)

is found to hold in all cases. These important theorems may be proved
independently of Huygens's construction by simply observing that, at
each point of the surface of separation, there must be a certain
connexion between the disturbances existing in the incident, the
reflected, and the refracted waves, and that, therefore, the lines of
intersection of the surface with the positions of an incident
wave-front, succeeding each other at equal intervals of time dt, must
coincide with the lines in which the surface is intersected by a
similar series of reflected or refracted wave-fronts.

In the case of isotropic media, the ratio (7) is constant, so that we
are led to the law of Snellius, the index of refraction being given by

µ = v1/v2 (8)

(cf. equation 1).

9. _General Theorems on Rays, deduced from Huygens's
Construction._--(a) Let A and B be two points arbitrarily chosen in a
system of transparent bodies, ds an element of a line drawn from A to
B, u the velocity of a ray of light coinciding with ds. Then the
integral [int]u^(-1) ds, which represents the time required for a
motion along the line with the velocity u, is a minimum for the course
actually taken by a ray of light (unless A and B be too far apart).
This is the "principle of least time" first formulated by Pierre de
Fermat for the case of two isotropic substances. It shows that the
course of a ray of light can always be inverted.

(b) Rays of light starting in all directions from a point A and
travelling onward for a definite length of time, reach a surface
[sigma], whose tangent plane at a point B is conjugate, in the medium
surrounding B, with the last element of the ray AB.

(c) If all rays issuing from A are concentrated at a point B, the
integral [int]u^(-1) ds has the same value for each of them.

(d) In case (b) the variation of the integral caused by an infinitely
small displacement q of B, the point A remaining fixed, is given by
[delta][int]u^(-1) ds = q cos [theta]/v_B. Here [theta] is the angle
between the displacement q and the normal to the surface [sigma], in
the direction of propagation, v_B the velocity of a plane wave
tangent to this surface.

In the case of isotropic bodies, for which the relation (8) holds, we
recover the theorems concerning the integral [int]µds which we have
deduced from the emission theory (§ 5).

10. _Further General Theorems._--(a) Let V1 and V2 be two planes in a
system of isotropic bodies, let rectangular axes of coordinates be
chosen in each of these planes, and let x1, y1 be the coordinates of a
point A in V1, and x2, y2 those of a point B in V2. The integral
[int]µds, taken for the ray between A and B, is a function of x1, y1,
x2, y2 and, if [xi]1 denotes either x1 or y1, and [xi]2 either x2 or
y2, we shall have
_ _
[dP]² / [dP]² /
------------------- | µ ds = ------------------- | µ ds.
[dP][xi]1 [dP][xi]2 _/ [dP][xi]2 [dP][xi]1 _/

On both sides of this equation the first differentiation may be
performed by means of the formula (3). The second differentiation
admits of a geometrical interpretation, and the formula may finally be
employed for proving the following theorem:

Let [omega]1 be the solid angle of an infinitely thin pencil of rays
issuing from A and intersecting the plane V2 in an element [sigma]2 at
the point B. Similarly, let [omega]2 be the solid angle of a pencil
starting from B and falling on the element [sigma]1 of the plane V1 at
the point A. Then, denoting by µ1 and µ2 the indices of refraction of
the matter at the points A and B, by [theta]1 and [theta]2 the sharp
angles which the ray AB at its extremities makes with the normals to
V1 and V2, we have

(µ1)² [sigma]1 [omega]1 cos [theta]1 =
(µ2)² [sigma]2 [omega]2 cos [theta]2.

(b) There is a second theorem that is expressed by exactly the same
formula, if we understand by [sigma]1 and [sigma]2 elements of surface
that are related to each other as an object and its optical image--by
[omega]1, [omega]2 the infinitely small openings, at the beginning and
the end of its course, of a pencil of rays issuing from a point A of
[sigma]1 and coming together at the corresponding point B of [sigma]2,
and by [theta]1, [theta]2 the sharp angles which one of the rays makes
with the normals to [sigma]1 and [sigma]2. The proof may be based upon
the first theorem. It suffices to consider the section [sigma] of the
pencil by some intermediate plane, and a bundle of rays starting from
the points of [sigma]1 and reaching those of [sigma]2 after having all
passed through a point of that section [sigma].

(c) If in the last theorem the system of bodies is symmetrical around
the straight line AB, we can take for [sigma]1 and [sigma]2 circular
planes having AB as axis. Let h1 and h2 be the radii of these circles,
i.e. the linear dimensions of an object and its image, [epsilon]1 and
[epsilon]2 the infinitely small angles which a ray R going from A to B
makes with the axis at these points. Then the above formula gives
µ1h1[epsilon]1 = µ2h2[epsilon]2, a relation that was proved, for the
particular case µ1 = µ2 by Huygens and Lagrange. It is still more
valuable if one distinguishes by the algebraic sign of h2 whether the
image is direct or inverted, and by that of [epsilon]2 whether the ray
R on leaving A and on reaching B lies on opposite sides of the axis or
on the same side.

The above theorems are of much service in the theory of optical
instruments and in the general theory of radiation.

11. _Phenomena of Interference and Diffraction._--The impulses or motions which a luminous body sends forth through the universal medium or aether, were considered by Huygens as being without any regular succession; he neither speaks of vibrations, nor of the physical cause of the colours. The idea that monochromatic light consists of a succession of simple harmonic vibrations like those represented by the equation (5), and that the sensation of colour depends on the frequency, is due to Thomas Young[11] and Fresnel,[12] who explained the phenomena of interference on this assumption combined with the principle of super-position. In doing so they were also enabled to determine the wave-length, ranging from 0.000076 cm. at the red end of the spectrum to 0.000039 cm. for the extreme violet and, by means of the formula (6), the number of vibrations per second. Later investigations have shown that the infra-red rays as well as the ultra-violet ones are of the same physical nature as the luminous rays, differing from these only by the greater or smaller length of their waves. The wave-length amounts to 0.006 cm. for the least refrangible infra-red, and is as small as 0.00001 cm. for the extreme ultra-violet.

Another important part of Fresnel's work is his treatment of diffraction on the basis of Huygens's principle. If, for example, light falls on a screen with a narrow slit, each point of the slit is regarded as a new centre of vibration, and the intensity at any point behind the screen is found by compounding with each other the disturbances coming from all these points, due account being taken of the phases with which they come together (see DIFFRACTION; INTERFERENCE).

12. _Results of Later Mathematical Theory._--Though the theory of diffraction developed by Fresnel, and by other physicists who worked on the same lines, shows a most beautiful agreement with observed facts, yet its foundation, Huygens's principle, cannot, in its original elementary form, be deemed quite satisfactory. The general validity of the results has, however, been confirmed by the researches of those mathematicians (Siméon Denis Poisson, Augustin Louis Cauchy, Sir G. G. Stokes, Gustav Robert Kirchhoff) who investigated the propagation of vibrations in a more rigorous manner. Kirchhoff[13] showed that the disturbance at any point of the aether inside a closed surface which contains no ponderable matter can be represented as made up of a large number of parts, each of which depends upon the state of things at one point of the surface. This result, the modern form of Huygens's principle, can be extended to a system of bodies of any kind, the only restriction being that the source of light be not surrounded by the surface. Certain causes capable of producing vibrations can be imagined to be distributed all over this latter, in such a way that the disturbances to which they give rise in the enclosed space are exactly those which are brought about by the real source of light.[14] Another interesting result that has been verified by experiment is that, whenever rays of light pass through a focus, the phase undergoes a change of half a period. It must be added that the results alluded to in the above, though generally presented in the terms of some particular form of the wave theory, often apply to other forms as well.

13. _Rays of Light._--In working out the theory of diffraction it is possible to state exactly in what sense light may be said to travel in straight lines. Behind an opening _whose width is very large in comparison with the wave-length_ the limits between the illuminated and the dark parts of space are approximately determined by rays passing along the borders.